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Questions tagged [dual-spaces]

The dual space of a vector space $V$ over a field $k$ is the vector space of all linear maps from $V$ into $k$.

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$T$ compact operator, Let $\Delta^*_{\bar{\lambda}}$ subspace of $X^*$. Prove $\Delta^*_{\bar{\lambda}} = \bar{\Delta^*_{\bar{\lambda}}}$

We have proved the following claim: Let T be compact operator and $\lambda \neq 0$. Then $\Delta_\lambda = \bar{\Delta _\lambda}$. Now there is the corollary: $T$ compact operator, Let $\Delta^*_{\...
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Dual basis of proper subspace $W$ of $\mathbb{R}^n$

Let $W\subset\mathbb{R}^n$ be a proper $r$-dimensional subspace of $\mathbb{R}^n$, and let the columns of the following matrix provide a basis for it: \begin{equation} F = \begin{bmatrix} | & &...
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How do we know the dual pairing between Lp spaces is well defined? [closed]

Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and let $X \in L^p(\Omega, \mathcal{A}, \mu)$ and $Y\in L^q(\Omega, \mathcal{A}, \mu)$. Then the dual pair betweent these spaces is defined as $\...
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Understanding spaces of negative regularity

Let $C^k(\mathbb{R}^n$) be the space of functions with $k$ continuous derivatives, and $H^s(\mathbb{R}^n)$ the Sobolev space $W^{2,s}$. Their dual spaces are commonly denoted as $C^{-k}$ or $H^{-s}$. ...
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Basis and dual basis in finite dimensional vector spaces

In finite dimension, a basis $\{f_i\}\in V$ and a dual basis $\{g_i\}\in V^*$ should satisfies $\langle f_i, g_j \rangle = \delta_{ij}$. I am wondering what is the terminology, if now I have a basis $(...
Yujie Zhang's user avatar
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Loomis and Sternberg Chapter 2, problem 2.19: defining the degree of a polynomial on a vector space (over R)

Exercise 2.19 of chapter 2 of L&S is: A polynomial on a vector space V is a real-valued function on V which can be represented as a finite sum of finite products of linear functionals. Define the ...
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Dual basis of a matrix space

I am considering the following basis $B$ of $S_2$: $$B=\left\{\begin{pmatrix}1&0\\0&0\end{pmatrix},\begin{pmatrix}0&1\\1&0\end{pmatrix},\begin{pmatrix}0&0\\0&1\end{pmatrix}\...
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If $f : U \rightarrow X$ is an isometric inclusion of Banach spaces, does $f' : X' \rightarrow U'$ have a bounded generalized inverse?

Let $X$ be a Banach space and let $U$ be a closed Banach subspace. The inclusion mapping $$ f : U \rightarrow X $$ induces a dual mapping $$ f' : X' \rightarrow U' $$ I am wondering about the ...
shuhalo's user avatar
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If $ \phi: X \to X^* $ is an isometry, then $ X $ is a complete space.

I am wondering if the following statement might hold (as I wanted to use this in solving another problem): If $ \phi: X \to X^* $ is an isometry, then $ X $ is a complete space. I know that $ X^* $ is ...
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I want to prove that $T$ belongs to $B(X)$, meaning it is a bounded linear operator.

Let $T_f: X \rightarrow X$ be defined by $T_f(x) = f(x) u$ for every $f \in X^*$ for some non-zero $u \in X$. I want to prove that $T$ belongs to $B(X)$, meaning it is a bounded linear operator. I ...
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Origin of notation of a dual space

Classically, there’s discordance in the notation of the dual of a vector space over a field $K$. Of course we can all agree that the algebraic dual is the vector space of linear maps from $V$ to its ...
Leonardo Lovera's user avatar
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Proving a linear functional belongs to $L^{4/3}(0,T; H^{-1})$

Let $H^{-1}$ be the dual to the Sobolev space $H^1$ and $T > 0$. Let $u: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $u \in L^\infty(0,T; L^2) \cap L^2(0,T; H^1)$. It can be shown that $$(u \...
CBBAM's user avatar
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why quojections are prequojections [closed]

While reading about quojections and prequojections in the book "Advances in the Theory of Fréchet Spaces," I'm having trouble understanding why every quojection is necessarily a ...
Assalami Med's user avatar
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Prove that the canonical mapping from an infinite dimensional vector space to it's double dual is a one-to-one mapping. [closed]

What is the canonical correspondence from a vector space V to it's double dual $V^{**}.$ Prove that this correspondence is one-one.($V$ need not be finite dimensional) I tried solving the problem in ...
Thomas Finley's user avatar
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1 answer
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Weak Star Convergence of Integral Averages

Suppose $U$ is a Banach space and let $V\subseteq U$ be bounded, convex and (norm-)closed. Consider the Bochner-Lebesgue space $L^r(0,T;U)$ with $T>0$ and $r\in[1,\infty]$ consisting of strongly ...
Joe S's user avatar
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Spanning set of support functionals in dual space

I am currently studying about supporting hyperplane (or, support functional) in dual space. Since, I am new in these topics I met with the following queries: Let $X$ be a normed space and $X^*$ be the ...
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(Copy) Set of linear functionals span the dual space iff intersection of their kernels is {0} .

I have fully understood the following question and got a motivation from it. Set of linear functionals span the dual space iff intersection of their kernels is $\{0\}$. My question is what will be the ...
Tutun's user avatar
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Let $A : X \to X$ be a linear operator such that $T_f A \in \mathcal{B}(X)$ for every $f \in X^*$. Show that $A \in \mathcal{B}(X)$.

Let $X$ be a Banach space and let $0 \neq u \in X$. (a) For every $f \in X^*$, let $T_f : X \to X$ be an operator defined by the prescription $T_f x = f(x) u$. Show that $T_f \in \mathcal{B}(X)$. (b) ...
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Assume that for every $f\in X^*$, there exists $y \in X$ such that $f(x)=\langle x, y \rangle$ for every $x \in X$. Show that $X$ is a complete space.

Let $(X, \langle \cdot, \cdot \rangle)$ be a real or complex vector space with an inner product. Assume that for every $f \in X^*$, there exists $y \in X$ such that $f(x) = \langle x, y \rangle$ for ...
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No infinite component in $\mathbb Z^2$ percolation implies an open circuit in the dual graph around the origin

I'm trying to make rigorous a geometric argument that seems extremely intuitive, but that I'm not managing to fully formalise. I'm considering percolation on $\mathbb Z^2$, and whether the origin ...
George's user avatar
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Let $X$ be a reflexive space and let $f \in X^*$, $ Px = x - \frac{f(x)}{\|f\|} y, \quad x \in X. $

Let $X$ be a reflexive space and let $f \in X^*$. (a) Show that there exists $y \in X$ such that $x - \frac{f(x)}{\|f\|} y \in \ker f$ for every $x \in X$. (b) Let $P : X \to X$ be the mapping defined ...
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4 answers
572 views

Dual space isomorphism non-canonical choice example

In a lot of resources that I have read it is mentioned that the isomorphism between $V$ and $V^*$ is non-canonical, but I was never sure that I properly understood precisely what this means. I haven't ...
lightxbulb's user avatar
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Confusion about Riesz representation theorem in $L^p$

Riesz-Representation Theorem (RR): let $(H,\langle\cdot,\cdot\rangle)$ be a Hilbert space. Then for every $T\in H'$, there exists a unique $v_T\in H$ s.t. $$T(u)=\langle u,v_T \rangle\quad \forall u\...
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Does $\#\text{Ker}(f^*)\le \#\text{Im}(f)$ hold? Duality of profinite group

Let $M, M'$ be a profinite groups. Let $M^*=\text{Hom}_{conti}(M,\Bbb{Q}/\Bbb{Z})$ be a dual of $M$. Let $f:M\to M'$ be a homomorphism of abelian group. Let $f^*: M'^*\to M^*$ be a map defined by $g\...
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What is the order of $\text{Ker}(\hat{f})$ when order of annihilator of $\text{Ker}f$ is given?

Let $A,B$ be an abelian group. $\hat{A}=\text{Hom}(A,\Bbb{Q}/\Bbb{Z})$,. Let $f$ be a map $ f : A\to B$, and $\hat{f}: \hat{B}\to \hat{A}$ be dual of $f$. In general, $\mathrm{Im}(f)$ is the ...
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Does the dual of a quiver representation correspond to the dual of the associated $\mathbb{K}Q$-module?

So, this question comes from the theory of quiver representations (more concretely, from the book "An introduction to Quiver Representations" by Harm Derksen and Jerzy Weyman). Let me lay ...
Duarte Costa's user avatar
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1 answer
129 views

Can the general idea of duality be made rigorous?

In mathematics and physics, there are many notions of duality that aren't always similar. For example, the notion of the dual of a vector space seems wildly different from the Hodge dual of linear ...
Níckolas Alves's user avatar
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Natural isomorphism between dual spaces

I'm having trouble following an interesting explanation of why the isomorphism between a vector space and its double dual is natural. The explanation is in an answer to a question posted on ...
Tomek Dobrzynski's user avatar
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Banach dual of the Hilbert-Schmidt operator

Let $T: L_2[0,1] \rightarrow L_2[0,1]$ be the Hilbert-Schmidt operator corresponding to the kernel $K(x,y)$. That is, $$(Tf)(x) = \int_0^1 K(x,y)f(y) dy$$ Then, for the Banach duel $T^\star: L_2[0,1]^\...
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For a finite dimensional vector space $V$, how does a basis of $V^*$ induce a basis of $V$?

Of course, given a basis of $V$, we can construct a dual basis for $V^*$ in the finite dimensional setting. Does the other direction hold in some capacity? For context, the sentence straddling pages ...
田中之夢's user avatar
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How do covectors and their basis work in Clifford/geometric algebra? [closed]

I'm curious about how covectors in linear algebra operate in Clifford/geometric algebra. So far, all the information I've come across regarding "dual vectors" seems to be connected to either ...
Mammal's user avatar
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Best way to think of the space $C^w((0,T); L^p(\mathbb{R})$

Let $f \in C^w((0,T); L^p(\mathbb{R}))$, i.e. $$\{f \mid f:(0,T) \rightarrow L^p(\mathbb{R}) \textrm{ is weakly continuous}\}.$$ Clearly this means for fixed $t$, $f(t,\cdot) \in L^p(\mathbb{R})$. I ...
CBBAM's user avatar
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1 answer
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The specific form of isomorphism from linear space $V$ to dual space $V*$

For a linear space $V$, we know that it is isomorphic to the dual space $V^*$, because they are both $n$-dimensional linear spaces. But what exactly this isomorphic mapping looks like seems to be ...
Daeree's user avatar
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1 answer
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Operator on reduced group $C^*$-algebra induces operator on von Neumann algebra

Let $\Gamma$ be a discrete group. Consider its reduced group $C^* $-algebra $C_\lambda^* (\Gamma)$ and von Neumann algebra $L(\Gamma) = C_\lambda^* (\Gamma)'' \subseteq B(\ell^2(\Gamma))$. Let $T:C_\...
Tomás Pacheco's user avatar
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1 answer
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Does there exist a linear map from $\ell^1$ to its bi-dual which is isometrically isomorphic?

I know that the space $\ell^1$ is not reflexive as the dual space $\ell^\infty$ is not separable but I don't know the dual space of $\ell^\infty$. Someone, please provide me with the details of the ...
Rudra's user avatar
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Dual of an inner product space (not necessarily Hilbert space) is Hilbert space?

I know that if H is an Hilbert space, then H' is also a Hilbert space. Now if X is just an inner product space (specifically not a Hilbert space), does it follow that X' is Hilbert? And if it is not ...
dip's user avatar
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Injectivity of the canonical map to the double dual

Let $V$ be an arbitrary vector space (possibly infinite-dimensional) and $V^{**}$ be its double dual. Define $$J: V \rightarrow V^{**}\\ \hspace{8mm} x \rightarrow \phi(x),$$ where $\phi(x): V^{*} \...
huh's user avatar
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Is the covariant structure of a covector with respect to coordinate changes forced, or is it a consequence of its definition as a linear functional?

In "The Geometry of Physics, an Introduction" (Theodore Frankel), we get the usual definition of a covector. Nominally, on page xxxii we get that From Frankel: $du^i$ is the linear ...
Nate's user avatar
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Clarification of estimates of $H^1$ norm and $H^{-1}$ norm.

Consider the following Dirichlet Problem for $u' \in H^1(\Omega)$ \begin{equation} \begin{cases} Lu' = f, \qquad \mbox{in } \quad\Omega \qquad (1)\\ u' = 0, \qquad \mbox{on } \quad \partial \Omega, \...
Jason Curran's user avatar
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How to prove the question about dual pair of Banach space?

Definition: Two Banach spaces $X$ and $Y$ form a dual pair $\langle X,Y\rangle$ if there is a bilinear or a sesquilinear form $\langle \cdot ,\cdot \rangle$ on $X\times Y$ which satisfies the ...
accretive's user avatar
1 vote
1 answer
63 views

How Can I Improve this Proof of Surjectivity between $V$ and $(V^\ast)^\ast$

Question I want to prove that $V$ is naturally isomorphic to $(V^\ast)^\ast$ without making reference to a basis. Note: $V$ is finite dimensional space. Attempt I have already shown that the map ...
Mr Prof's user avatar
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1 answer
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Exercise 3.F.29(a) in "Linear Algebra Done Right 4th Edition" by Sheldon Axler.

Exercise 3.F.29(a) Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal{L}(V,W)$. (a) Prove that if $\varphi \in W'$ and $\text{null} \ T' = \text{span} \ (\varphi)$, then $\text{range} \ T =...
Paul Ash's user avatar
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2 votes
1 answer
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$S \in \mathcal{L}(L^1(\Omega))$, find $T^* \in \mathcal{L}(L^\infty(\Omega))$ with $T^*g = Sg \forall g \in L^1(\Omega) \cap L^\infty(\Omega)$

Below I will bring a passage from Heat Kernels by Wolfgang Arendt (Theorem 4.3.3, page 52). I need to understand it and write a more verbose report based on the chapter, however I am stuck at this ...
Meta-chan's user avatar
4 votes
1 answer
164 views

dual space of l2 with strange norm

Consider $\displaystyle(\ell_2, \lVert\cdot\rVert_\star), \lVert x\rVert_\star = \sum\limits_{k=1}^{\infty}\frac{|x(k)|}{k}$. What is its dual space? Is this space reflexive? My idea is to consider $\...
GeoArt's user avatar
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0 answers
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Covectors, Metric Tensors and Unit Vectors [closed]

In orthogonal cartesian coordinates $c\cdot c=|c|^2$ is the square of length of the vector $c$. In oblique coordinates the square of the length of $c = c * g * c$ where g is the metric tensor and $g*c$...
R. Emery's user avatar
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3 votes
1 answer
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The dual $(L^\infty)^{*}$ is not $L^1$ by constructing example

The problem statement is the same as this post: $L^{\infty *}$ is not isomorphic to $L^1$ . Let $L^\infty = L^\infty(m)$, where $m$ is Lebesgue measure on $I=[0,1]$ . Show that there is a bounded ...
Nazono Sumiko's user avatar
4 votes
1 answer
78 views

How do we know the space of distributions is "big"?

Consider the set of compactly supported smooth functions $C^\infty_c(\mathbb{R})$. The dual of this space, $(C^\infty_c(\mathbb{R}))'$ is often referred to as a very large space. Since every $f \in C^\...
CBBAM's user avatar
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1 vote
1 answer
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Can $C_p^\infty(\mathbb{R}^n)$ be considered as a dual space of $T_p\mathbb{R}^n$, where $p\in \mathbb{R}^n$?

warning: This may be a stupid question, because I am poor at differential euqations $v_p\in T_p\mathbb{R}^n$ is a linear function over $C_p^\infty (\mathbb{R}^n)$, where $C_p^\infty (\mathbb{R}^n)$ ...
Richard Mahler's user avatar
1 vote
1 answer
76 views

Why does there exist unique numbers $\varphi_1(v), \ldots, \varphi_m(v)$ such that $ Tv = \varphi_1(v)w_1 + \cdots + \varphi_m(v)w_m$?

Exercise. Suppose $T \in \mathcal{L}(V,W)$ and $w_1,\ldots,w_m$ is a basis of $\text{range} \ T$. Hence for each $v \in V$, there exist unique numbers $\varphi_1(v), \ldots, \varphi_m(v)$ such that $$ ...
Paul Ash's user avatar
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3 votes
2 answers
53 views

Dual space of $L^p([0,1])$ for $0 < p < 1$.

I convinced myself that in the space $L^p([0,1])$ there is no other non-empty, open, convex subset different from $L^p([0,1])$ itself when $0 < p < 1$. I was told that I can conclude from this ...
Mads C's user avatar
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